I like poker and I like quantum computing and lo and behold here is a paper with both:

arXiv: 0902.2196Title: Quantized PokerAuthors: Steven A. Bleiler

Poker has become a popular pastime all over the world. At any given moment one can find tens, if not hundreds, of thousands of players playing poker via their computers on the major on-line gaming sites. Indeed, according to the Vancouver, B.C. based pokerpulse.com estimates, more than 190 million US dollars daily is bet in on-line poker rooms. But communication and computation are changing as the relentless application of Moore’s Law brings computation and information into the quantum realm. The quantum theory of games concerns the behavior of classical games when played in the coming quantum computing environment or when played with quantum information. In almost all cases, the “quantized” versions of these games afford many new strategic options to the players. The study of so-called quantum games is quite new, arising from a seminal paper of D. Meyer \cite{Meyer} published in Physics Review Letters in 1999. The ensuing near decade has seen an explosion of contributions and controversy over what exactly a quantized game really is and if there is indeed anything new for game theory. With the settling of some of these controversies \cite{Bleiler}, it is now possible to fully analyze some of the basic endgame models from the game theory of Poker and predict with confidence just how the optimal play of Poker will change when played in the coming quantum computation environment. The analysis here shows that for certain players, “entangled” poker will allow results that outperform those available to players “in real life”.

Comments

The ensuing near decade has seen an explosion of contributions and controversy over what exactly a quantized game really is and if there is indeed anything new for game theory.

It gives you a manifest model for Harsanyi’s ideas about mixed-strategy equilibria.

… it is now possible to fully analyze some of the basic endgame models from the game theory of Poker and predict with confidence just how the optimal play of Poker will change when played in the coming quantum computation environment.

I think he’s an optimist. I don’t think the environment is going to change anything. I do think that optimal solutions/best-response strategies are going to have a lot of entanglement, and be highly degenerate.

Good players are going to come up with semantic descriptions that let them distinguish between ‘different-from-me’ and ‘better-or-worse-than-me’ [because most players are so arrogant that they equate the two].

I also think that thermodynamics is hand in hand with the quantum here.

Note that weird things happen if the players can be quantum entangled. Really. New Nash equilibria suddenly appear…

Likewise, if the referee and player(s) are entangled.

My coauthor Phil Fellman and I discussed this face-to-fase with John Forbes Nash, Jr., who agrees. And admits that he hadn’t possibly imagined that in his early 1950s breakthrough, now seen as far more robust than originally.

Nor are these “toy problems.” Future geopolitical conflicts will involve States (and non-state players) who have quantum computing and quantum communications. Hence this becomes a life or death matter.

Let me repeat three arguments against quantum games in general, and quantum poker in particular:

1) physics, as far as I know, does not prescribe how to canonically quantize poker. Every quantum game is an ad-hoc construction. A quantum game does not teach us anything about quantum physics.

2) if you change the rules of an old game, you get a new game. The fact that new Nash equilibria appear in the new game is no surprise and does not imply anything about the old game.

3) by including the new “quantum” moves in a pay-off matrix, one can reformulate the quantum game as a purely classical game (=old game plus some new classical moves plusw their consequences). Quantum games teach us nothing about quantum mechanics.

Let me illustrate some of these points: suppose we add the following rule to poker: if you weigh more than 87 kg, your flush is lower than someone else’s two pairs, unless that person weighs less than 67 kg.

Good, now we can call this “gravitized poker” and wonder about new Nash equilibria. And it turns out, if you have a flush, and you weigh more than 87 kg, you should change your betting strategy when up against an opponent who weighs more than 67 kg.

Note that weird things happen if the players can be quantum entangled.

This is really one of the things that tweaks me about classical game theory. The latter makes the very brash assumption that information is an externality [and sometimes even worse, that the information rate is 0]. Formulating the game in quantum terms makes the information intrinsic to the structure of the game. [That’s what I meant by Harsanyi made manifest].

So, you know, some more things that have brewed through. Quantum information is a nice model for the game, but I don’t know that quantum processing solves it any better. Specifically, poker is a game on a directed graph. So there are no closed loops, and consequently [and despite what other people may say] its not a potential game [a la Monderer and Shapley1]. You don’t necessarily get nice eigenfunction state spaces.

Even if we tied the end of the game off at ‘infinity’ and wrapped it around to the start [and formulated it as such as a learning problem], we still need to resolve that there are two communications channels [cards and bets]. Learning has a cost and it isn’t a just-move-the-zero-of-your-potential cost. It is functional, and directly related to how you interact with the information. [Right now, I think most people have some mental formulation of a mean-field-theory of their opponent’s play].

1 This paper has been completely abused by the network systems people. Its a first start at a discrete analogue for a lot of things that are taken for granted in physics. In this formulation, information rate is infinite and communication is ephemeral. Identifying ‘packets’ and ‘interactions’ [as a typical particle physicist might do] is completely ignored. This tweaks me because that is really the heart of mechanism design. Maybe I’m just an aesthetic twat.

However, what if changing the rules included randomizing over the strategies? If so, then any Nash equilibria that arise using mixed strategies must not imply anything about the old game either! This will disturb many game theorists.

“by including the new “quantum” moves in a pay-off matrix, one can reformulate the quantum game as a purely classical game (=old game plus some new classical moves plusw their consequences). Quantum games teach us nothing about quantum mechanics.”

I cannot comment on whether quantum games teach us anything new about quantum mechanics or not; however, I want to point out that the first part of this argument is similar to the argument by van Enk and Pike in “Classical rules in quantum games” 2002. Arguments like these are made without regard to a game theoretical context and ignore a fundamental goal of game theory, which can be informally stated as: creating new opportunities with positive results for everyone (isn’t this what all societies have been striving for since time immemorial?). This is why mixed strategies are introduced in games, as well as mediated communication, for example. Quantization is just another venue by which new opportunities for the players might arise, ideally in the form of a Nash equilibrium. A new opportunity that player’s explore the existence of in Prisoner’s Dilemma is to escape the dilemma. This is impossible using mixed strategies and mediated communication, but as Eisert et al show in “Quantum Games and Quantum Strategies”, 1999, is possible via a suitable quantization of the game. The mathematical context that addresses such arguments is one which veiws the normal form of a game as a function, the domain of which is to be extended such that new opportunities relevant to the players might potentially be realized. Such a context is discussed in section 4 of Bleiler’s quantized poker paper, as well as his paper found here: http://arxiv.org/abs/0808.1389.

Allowing mixed strategies does not change the rules of the game by any *reasonable* definition (in other words, I find Bleiler’s definition far from reasonable). In poker, making *random* decisions about betting or folding in a given situation, rather than making always the same decision in that same situation, is independent of (or not part of) the rules the casino enforces. The casino cannot in any reasonable way enforce players to use only pure strategies.

On the other hand, the casino, as well as the other “classical” players, would protest if certain players at the poker table would start to generate entangled pairs of qubits between them, and then perform measurements in order to make decisions.

About the Prisoner’s Dilemma: the players can avoid the dilemma if the rules of the game change. That is rather trivial, but that is what happens by allowing the players to share entangled state and have a helpful interrogator to make certain measurements on the final state. With a helpful interrogator, even a classical one, there wouldn’t be a dilemma in the first place.

That some of my arguments sound similar to those of Van Enk and Pike is indeed remarkable.

Steven writes “About the Prisoner’s Dilemma: the players can avoid the dilemma if the rules of the game change.”

This is of course true. I think this confusion certainly arises if a quantum game is defined to be just a game that is played using quantum rules and teh corresponding classical game is embedded within in one possible way or the other.

However, one can aviod this by first tranforming a classical game into a suitable format that permits establishing a one-to-one correspondence between a classical game and the “classicality” of the physical system being used to play the game.

Successfully establishing this correspondence makes possible to have almost a natural extension of a classical game when the “classicality” assumption is dropped in the transformed version of the game.

I am not sure if the resulting quantization will be cannonical. However, it seems that it will be a very much convincing one.

Whether a quantum game teaches us anything about quantum physics is an open question, which I believe is not at the centre of the area of quantum games.

I think the question of interest to quantum games is whether quantum mechanics permits a reasonable extension of game theory to the quantum domain. To me it seems that the answer to this qusetion is yes.

Steven writes: “by including the new “quantum” moves in a pay-off matrix, one can reformulate the quantum game as a purely classical game (=old game plus some new classical moves plusw their consequences). Quantum games teach us nothing about quantum mechanics.”

Re:
This is the old argument often referred to Enk and Pike.

One can of course construct some classical model of a quantum game. However, this does not mean that the quantum game is nothing beyond the corresponding classical game.

This argument is same as to say that as one can describe a Hydrogen atom in some (arbitrary) classical way, therefore, the quantum model is in fact classical.

“Quantum games teach us nothing about quantum mechanics.”

Why this is important? A quantum game offers a reasonable way to extend a classical game towards quantum domain. Why such an extension is supposed to open new avenues for quantum mechanics? It is the game theory for which a new avenue is opened.

Steven writes: “if you change the rules of an old game, you get a new game. The fact that new Nash equilibria appear in the new game is no surprise and does not imply anything about the old game.”

The process of tranforming a game into a suitable form, such that classicality of a shared physical system is placed in a one-to-one correspondence with the classical game and its particular outcome, is different from “changing the rules of the game.”

If such a correspondence is successfully established then the corresponding quantum game emerges naturally by dropping the classicality assumption.

The rules of the game remain identical in the transformed version of the classical game and the quantum game.

The tranformed game may give the impression of having different rules, which should be seen in conjunction with the classicality of the shared physical system.

I refer to classicality of a shared physical system, used in the physical implementation of game, in view of a relevant set of Bell inequalities.

Re:
There is a long history of several ad-hoc constructions offered as (classical) models of the Hydrogen atom before constructing a right quantum mechanical model became possible.

How to quantize a game in a convincing way is certainly not an easy question and one must accept that this is going to be achieved after several (incorrect) attempts. This, however, does not mean that th whole exercise is futile.

Azhar I don’t understand your point of view concerning going between quantum and classical games. Repeatedly you make it sound as if there is a “unique” way to go to quantum theory from a classical game. Ignoring that there are many ways to quantize a classical field theory, for example, it seems to me that game theory is about information resources, and in this context “classical limits” are notoriously hard to define. This is more closely tied to the arise of “classical information” than to your examples in the hydrogen atom. For example there are numerous quantum models which give rise to the same “classical limit.” Which quantum game is then relevant?

Well, I finally realize that Steven is none other than Van Enck himself. It is about time! Now the last sentence of his response to my response to his post (how Harsanyi-ish this sounds) on Feb 18, 2009 make complete sense. As they say on Facebook, Faisal is: Red-Faced.

Dave I regret if I have unintentionally made it to sound that there is a unique way to quantum game from a classical game. I definitely agree that there can be several convincing ways and that my suggestion proposes just one of those. However, I think that if the mentioned correspondence is truly one-to-one then it may possibly unify some of those several ways. The argument based on information resources and using classical to quantum information is certainly another nice and convincing way but not the only one. I believe your question asking which quantum game is relevant is connected to how we establish the mentioned correspondence.

“How to quantize a game in a convincing way is certainly not an easy question and one must accept that this is going to be achieved after several (incorrect) attempts. This, however, does not mean that th whole exercise is futile.”

When you say “quantizing a game in convincing way”, I presume that you have a mathematically formal game-theoretic context in mind; that is, a convincing quantization should be an extension of the normal form of the game the same way a mixed game is an extension. Bleiler calls such quantizations “proper” in his paper found at

In my humble opinion, constructing proper quantizations of games is a fundamental and non-trivial task in quantum game theory.

As for the incorrect attempts at quantization of games you mention, I think perhaps “non-convincing” is a better way to characterize them. Some of these attempts may not result in proper quantizations, and may therefore lack game-theoretic context. But as you said, that does not necessarily mean they are futile attempts. I think they likely carry scientific merit.

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